Reference

A function f between topological spaces is sequentially continuous if the image of every convergent sequence is a sequence which converges to the image of the limit. Continuity always implies sequential continuity: Suppose xn → x. Then if U is any open neighborhood of f(x), f − 1(U) is a neighborhood of x which (by continuity of f) is open.

A function : between two topological spaces X and Y is continuous if for every open set , the inverse image = {| ()} is ... For non-first-countable spaces, sequential continuity might be strictly weaker than continuity. (The spaces for which the two properties are equivalent are called sequential spaces.) This motivates the consideration of ...

We remark here that continuity defined in this way is also called sequential continuity. In metric spaces, sequential continuity is equivalent to continuity as described by the \(\epsilon - \delta \) definition given above (Simmons, 1983). But in more general spaces, which we describe later—topological spaces—this may not always be so.